Rectifiability of a class of integralgeometric measures and applications

TL;DR

This paper proves the rectifiability of a class of integral geometric measures for p > 1, resolving a long-standing problem and exploring implications for complex analysis and geometric measure theory.

Contribution

It resolves Federer’s open problem on rectifiability of integral geometric measures with p > 1 and extends classical theorems to new settings, impacting analytic capacity and removability.

Findings

Confirmed rectifiability of integral geometric measures for p > 1

Extended Vitushkin's conjecture results to multi-scale settings

Generalized Besicovitch-Federer projection theorem beyond sigma-finite sets

Abstract

We resolve a long-standing open problem posed by Federer concerning the rectifiability of the integral geometric measure with exponent p >1, thereby settling a question that has persisted since its formulation. While the main theorem is unchanged from previous versions, the exposition and applications have been substantially revised to highlight the result's consequences for Vitushkin's conjecture on analytic capacity and removability in the complex plane. As an application, we establish two novel results related to Vitushkin's conjecture: in a multi-scale setting, we provide an affirmative answer for sets with finite integral geometric measure within regimes of Favard length behavior at small scales not previously addressed; and in a single-scale framework, we extend the Besicovitch-Federer projection theorem beyond the classical sigma-finite setting, namely for planar sets intersecting a typical line in finitely many points. The rectifiability criterion for general Radon measures via slicing, included in earlier versions, has been removed and will appear in separate work.